Collective Excitation Dynamics of a Cold Atom Cloud

T. S. do Espirito Santo,1, 2 P. Weiss,3 A. Cipris,3 R. Kaiser,3 W. Guerin,3 R. Bachelard,3, 4 and J. Schachenmayer2

1Instituto de Fısica de Sao Carlos, Universidade de Sao Paulo - 13560-970 Sao Carlos, SP, Brazil2ISIS (UMR 7006) and IPCMS (UMR 7504), Universite de Strasbourg, CNRS, 67000 Strasbourg, France

3Universite Cote d’Azur, CNRS, Institut de Physique de Nice, France4Departamento de Fısica, Universidade Federal de Sao Carlos,

Rod. Washington Luıs, km 235 - SP-310, 13565-905 Sao Carlos, SP, Brazil

We study the time-dependent response of a cold atom cloud illuminated by a laser beam imme-diately after the light is switched on experimentally and theoretically. We show that cooperativeeffects, which have been previously investigated in the decay dynamics after the laser is switchedoff, also give rise to characteristic features in this configuration. In particular, we show that col-lective Rabi oscillations exhibit a superradiant damping. We first consider an experiment that isperformed in the linear-optics regime and well described by a linear coupled-dipole theory. We thenshow that this linear-optics model breaks down when increasing the saturation parameter, and thatthe experimental results are then well described by a nonlinear mean-field theory.

I. INTRODUCTION

The optical response of a coherently illuminated cloudof coupled scatterers can dramatically differ from thelight emission properties of its individual constituents.Such collective/cooperative effects have been intensivelyexplored in recent years, especially with cold atoms [1, 2].In particular, super- and sub-radiance have been recentlyinvestigated in various experimental geometries [3–9].Strikingly, the current experimental observations are wellexplained in the low-excitation limit [10], where dynam-ics can be described by linear equations of motions ofclassical coupled dipoles [11, 12]. It is an importanttask to explore collective effects beyond this linear-opticsregime [13–17].

In recent cold-atom experiments, super- and sub-radiance have been studied by observing the decay dy-namics after the driving laser is switched off [3–9]. Here,we demonstrate that the dynamics immediately after thelaser switch-on can also be used to observe cooperativeeffects. We show that in the linear-optics regime the dy-namics of the scattered light intensity can be modeledby that of an effective single driven dipole. By fittinga function for the evolution of the intensity emission ofthis effective dipole [18, 19], we can extract collective de-cay rates and frequency shifts. The cooperative shiftshave been recently understood in terms of a multi-modecollective vacuum Rabi splitting [19]. In this paper wewill focus on the collective damping rates and show thatthey are consistent with those of the experimental ob-servations in the superradiant regime of the switch-offdynamics [5].

While most experimental observations are consistentwith a linear-optics model in the low-saturation regime,in this paper we also consider the case of larger saturationparameters, and show that experimental signatures startto deviate from the linear model. For this situation, weshow that the observed switch-on dynamics can, however,be well described by a non-linear mean-field theory.

This paper is organized as follows: In Sec. II we present

an experiment-theory comparison for the linear-opticsregime. We analyze the switch-on dynamics theoreti-cally and show that the experimental data demonstratessuperradiance. In Sec. III we then proceed to show thatfor larger saturation parameters the nonlinear mean-fieldtheory provides a better model for the experimentallyobserved dynamics. Lastly, we conclude and provide anoutlook in Sec. IV.

II. SWITCH-ON DYNAMICS IN THELINEAR-OPTICS REGIME

In this section, we show how superradiance can be ob-served in an experiment monitoring the switch-on dy-namics of a cold-atom cloud in the linear-optics regime.We start by briefly describing the experimental setup(Sec. II A) and the linear optics model (Sec. II B). Wethen compare full numerical simulations to experimentaldata (Sec. II C) and show that the cloud dynamics canbe modeled by the dynamics of a single effective drivenand damped dipole (Sec. II D). We then demonstrate(Sec. II E) that the damping of the collective oscillationexhibits a superradiant rate, similar to the one observedin the switch-off dynamics [5].

A. Experimental setup

The experimental data discussed in this section wasobtained with the same setup as in [5]. A precise de-scription of the experiment can thus be found in thisreference.

In brief, we produce a three-dimensional Gaussiancloud (rms width R ≈ 1 mm) of N ≈ 109 randomly dis-tributed 87Rb atoms [see sketch in Fig. 1(a)]. The atomsbehave essentially as two-level systems, using the closedatomic F = 2 → F ′ = 3 transition (wavelength λ =2πc/ω0 = 780.24 nm and linewidth Γ/2π = 6.07 MHz).The cloud is homogeneously illuminated by a linearlypolarized monochromatic probe beam with beam waist

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0 2 4 60

0.5

1

1.5

2

0 2 4 60

0.5

1

1.5

2

(d)

0 2 4 60

0.5

1

1.5

2

(b) (c)

FIG. 1. (a) Sketch of the experiment: A probe beam (detuned by ∆ = ωL−ω0 from the atomic transition) suddenly illuminatesa Gaussian cloud of two-level atoms (transition linewidth Γ, optical depth b0). The time-dependent scattered light intensity ismeasured at an angle of θ = 35 from the axis of the incident beam. (b-d) Experimental time-dependent intensity response forvarious cloud and laser parameters showing damped Rabi oscillations. The damping depends on the optical depth b0 and ∆.Lines denote a full numerical simulation of linear-optics equations of motion (see text) and reproduce the experiment very welldue to the low saturation parameter (s ≈ 0.02).

w ≈ 5.7 mm, frequency ωL and detuning ∆ = ωL − ω0

from the atomic transition. It is propagating along thez-direction, k0 = k(0, 0, 1)T , k = 2π/λ.

Multiple series of pulses with 10%-90% rise time ofabout 6 ns, which is short compared to the lifetime ofthe excited state τat = Γ−1 = 26.2 ns, are produced byacousto- and electro-optical modulators used in series.During a cycle of pulses the atomic cloud expands balis-tically, which allows us to probe different on-resonancepeak optical depths. The optical depth is defined asb0 = σsc

∫ρ(0, 0, z)dz, with σsc the atomic cross-section.

Accounting for the internal structure of Rubidium, thiscorresponds to b0 = (7/15)3N/(kR)2 in the experiment.In the scalar-light model used below, the optical thicknessis given by bS0 = 2N/(kR)2. We also vary the detuningof the probe pulses but then adjust the light intensityaccordingly to keep a constant saturation parameter ofs ' (2.2 ± 0.6) × 10−2. The time-dependent scatteredlight intensity is recorded by a photon detector in the farfield at an angle of θ = 35 from the z-axis.

To clean the recorded intensity signal from remainingtechnical imperfections of the light switch-on dynamics,such as small overshoots, we divide the normalized tem-poral signal recorded with the atoms by a normalizedreference profile of the laser without atoms (and whitepaper as scattering medium). The experimental signal isaveraged over a large number of realizations (≈ 5 × 105

cycles), and normalized to one in the steady state.

B. Linear-optics dynamics

In the limit of low excitation in the cloud, i.e. in aregime where the atoms are only virtually excited andall population remains in the ground state (see Sec. IIIfor a detailed derivation), the dynamics is governed by

the well-known linear coupled-dipole (CD) equation

d

dtb(t) = Mb(t) + w. (1)

Here, the system is described by the vector of complexexcitation amplitudes, b(t) = [β1(t), β2(t), . . . , βN (t)]T .The laser excitation is governed by the Rabi-frequencyvector w = −i[Ω1, Ω2, . . . ,ΩN ]T /2 , where the complexΩn = Ω0eik0·rn contain the single-atom Rabi frequencyas well as the laser phase due to the random positions ofthe atoms, rn. Explicitly, the elements of the matrix Mare

Mnm = δnm

(i∆− Γ

2

)+ (δnm − 1)Gsnm. (2)

The diagonal term governs the single-atom dynamics,while the off-diagonal part includes all long-range dipole-dipole couplings between the atoms. In our setup weconsider a cloud of low density, with typical separationbetween two atoms n and m, rnm = |rn − rm| k−1.In particular, our experimental peak density of ρ0 ≈0.06λ−3 corresponds to a typical particle separation of

rnm = 2ρ−1/30 ∼ 30k−1. Given the large distance be-

tween neighbouring atoms the physics will be largelydominated by the dipole-dipole far-field terms. Such aregime is well described by a scalar-light model, for whichone finds [20]

Gsnm =Γ

2

eikrnm

ikrnm. (3)

The formal time-dependent solution of the problemwhen initially all atoms are in the ground state is

b(t) =[eMt − 1

]M−1w. (4)

The experimentally measured intensity is related tothe square of the electric far field at the detector

3

position. Defining the observation direction n =(sin θ cosφ, sin θ sinφ, cos θ), with angles defined forspherical coordinates with the incident laser wave vectork0 along the z axis, the intensity signal is proportionalto

I(θ, φ) =

N∑n,m

β∗nβmeikn·(rn−rm). (5)

In our numerical simulations we integrate the signal over

the azimuthal angle φ, Iθ ≡∫ 2π

0dφ I(θ, φ), to reduce

small-N fluctuations, and use θ = 35 as in the experi-mental setup. We always consider the normalized steady-state value Iθ(t→∞) ≡ Is = 1.

C. Numerical comparison

In Fig. 1(b-d) we compare the experimentally recordedtime-dependent response signal to a full numerical sim-ulation of the cloud. Since treating N = 109 atomsis out of reach numerically, even for the linear regime,we model the experiment with an effective cloud con-sisting of Neff = 5000 atoms and a rms radius chosento match the experimental optical thickness, i.e. R =k−1

√2Neff/b0. For the results in Fig. 1 we take the av-

erage over 100 realizations of random positions of theatoms. For numerical stability we exclude realizationswith two atoms much closer than the typical distance

between neighbors, here we chose dmin = 0.1ρ−1/30 with

ρ0 the peak density of the effective cloud.The three panels correspond to different experimental

situations with varying detuning and optical thickness:(b) b0 = 14.2 and ∆ = −8Γ; (c) b0 = 13.4 and ∆ = −3Γ;and (d) b0 = 46.3 and ∆ = −3Γ. The intensity evolutionis reproduced by the simulations of the linear model verywell, although the number of particles used is differentfrom that of the experiments by orders of magnitude.This highlights the central role of the resonant opticalthickness as control parameter of the collective dynamicsof the dipoles. In order to validate the simulations wechecked that results for different Neff (and thus differentdensities) as well as different dmin are indistinguishablefrom each other. We find the main difference to the ex-periment in the height of the first oscillation, which istypically lower in the experiment than in the simulation.We attribute this more damped behavior mainly to thefinite switch-on time for the laser.

The oscillations after switch-on are generally moredamped with increasing b0 and decreasing |∆|. This ise.g. seen by the decreased amount of visible oscillationswhen decreasing |∆| at constant b0 [going from panel (b)to panel (c)], and by the even increased damping whenkeeping ∆ constant and increasing b0 [going from panel(c) to panel (d)]. A systematic study of the damping asfunction of ∆ and b0 is discussed in Fig. 4 below.

Note that the experimental observation does not onlyexcellently agree with the CD simulations, but also with

a linear-dispersion theory [21]. This was demonstratedin [19], where we showed that a modification of the oscil-lation frequency can be very well understood in terms ofa multi-mode vacuum Rabi-splitting within this linear-dispersion theory framework.

D. Effective mode

Remarkably, we find that the intensity dynamics of thecloud (averaged over many experimental runs) approxi-mately resembles the evolution of a single driven super-radiant dipole. This implies that we can fit it well to aphenomenological function of the form [18, 19]

Iθ = Is

∣∣∣1− e(iΩN−ΓN/2)t∣∣∣2 , (6)

where ΓN and ΩN denote the decay rate and general-ized Rabi frequency of the effective mode, respectively.Three examples of fits are shown in Fig. 2(a-c) whereboth the experimental data, and the full numerical sim-ulations have been fitted using Eq. (6). The fit to CDsimulations becomes nearly perfect in the limit of low b0and large |∆| [Fig. 2(a)]. For larger b0 and smaller |∆|the fits are still remarkably good and allow us to extracteffective mode properties of the cloud. Similarly, we findthat most of the experimental data can be very well fit-ted by Eq. (6) as also shown in Fig. 2. In the followingwe discuss the effective mode picture in different limits.

1. Single-mode limit (timed Dicke regime)

The agreement of the single-mode fits to the linearcoupled-dipole theory in the limit |∆| Γ can be under-stood from the fact that the equilibration dynamics (onthe scale of tΓ ∼ 10) is dominated by a single macroscopicscattering mode, which has been e.g. understood in termsof the excitation of a timed Dicke state [10, 22]. Since thedipoles’ excitation is essentially determined by the laserin the regime of low optical thickness (b0 ∼ 0.1− 1), it isconvenient to move to the laser frame by defining

βn = e−ik0·rnβn. (7)

This leads to coupled dipole equations of the form

d

dtb(t) = Mb(t) + w, (8)

where the Rabi frequency vector is now homogeneous inphase, w = −i(Ω0/2)[1, 1, . . . , 1]T , and the coupling ma-trix M becomes

Mnm = δnm

(i∆− Γ

2

)+ (δnm − 1)Gsnmeik0·(rm−rn).

(9)

4

0 2 4

tΓ

0

1

2

I θ(a) b0 = 7.8, ∆ = −10Γ

exp.

CD

0 2 4 6

tΓ

0.0

0.5

1.0

1.5

(b) b0 = 18.9, ∆ = −5Γ

0 5 10

tΓ

0.0

0.5

1.0

(c) b0 = 42.8, ∆ = −1.5Γ

FIG. 2. Examples of an effective mode fitting [Eq. (6)] to signals from the experiment (points: data, solid lines: fit) and to CDsimulations (triangles: simulation, dashed lines: fit) in various regimes: In the under-damped case (a) the fit to the CD theoryis nearly perfect. In the more damped cases (b-c) small deviations from the effective mode (also in CD simulations) are visible.

Taking advantage of this uniform driving in the laserframe, we may replace the amplitude vector with identi-

cal amplitudes, b(t) ≡ β[1, 1, . . . , 1]T . This structure isenforced by taking β to evolve as the mean value of allindividual coherences, i.e., by summing Eq. (8) over theatoms (this implies that the sum of dipole-dipole interac-tions for each atom is approximated by its mean value),so one obtains

d

dtβ(t) =

[i∆− Γ

2− C

]β(t)− i

Ω0

2, (10)

with the following complex correction:

C =1

N

N∑n,m;n6=m

Gsnmeik0·(rm−rn). (11)

By definition, in this approximation the solution to (10)is given by that of a single dipole,

β(t) =[1− e(iΩsd

N−ΓsdN /2)t

] iΩ0

2iΩsdN − Γsd

N

, (12)

with modified frequency and damping

ΓsdN ≡ Γ + 2ReC, (13)

ΩsdN ≡ ∆− ImC. (14)

For the time-dependent intensity signal, this implies

I(t) = |β(t)|2N∑n,m

ei(kn−k0)·(rn−rm). (15)

In this single-mode limit, the intensity evolution is thusreproduced by a fitting function of the form of Eq. (6).The geometrical factor in Eq. (15), depending on par-ticle and detector position, only modifies the normal-ization factor, i.e. the steady-state intensity. The time-dependence of the intensity evolution is independent from

the measurement direction in the effective-mode approx-imation. Note that the geometrical factor features thecharacteristic ∝ N2 enhancement for an intensity mea-surement in the laser direction, when all terms in thedouble sum in Eq. (15) contribute coherently [23]. Off-axis measurements only lead to an intensity ∝ N , sincethe different random phases for n 6= m in the sum averageto zero. Note that this property is generally not foundto be true when experimentally studying the collectivefrequency, where a more elaborate theoretical analysis isnecessary [19].

To compute the value of C, we assume a Gaussiandistribution of atoms and replace the sums in Eq. (11)by an integral. The integration gives

C =ΓN

2(2π)3R6i

∫dr

∫dr′

eik|r−r′|

k|r− r′|eik0·(r−r′)e

−r2+r′2

2R2

= −Γ

2

bS08

[2iD(2kR)√

π− (1− e−4k2R2

)

](16)

Here, D(. . . ) denotes the Dawson integral, which asymp-totically behaves as D(x → ∞) ∼ 1/(2x). This leads toa shift in frequency of the single mode in dilute cloudsthat scales with the density, reminiscent of cooperativeLamb shifts [24]. In our dilute sample (kR ∼ 104 forthe experiment and kR & 10 for the simulations), theimaginary part of C is much smaller than the transitionnatural linewidth. Consequently, in our dilute clouds thisdensity-dependent shift cannot be seen. In contrast, opti-cal thickness-dependent shifts in the oscillation frequen-cies observed at the switch-on can be interpreted as amulti-mode vacuum Rabi splitting, and represent a mea-sure of the coupling strength of the light modes to theatomic cloud, as shown in [19]. Those shifts are not in-cluded in the simplistic single-mode limit.

As for the decay rate, the exponential term in Eq. (16)

5

is strongly suppressed in large clouds, so that

C =Γ

2

bS08. (17)

This implies that in the single-mode limit the dampingrate of the effective dipole would be expected to be

ΓsdN = Γ

(1 +

bS08

). (18)

Note that the single-mode assumption thus re-producesthe well-known result for the scaling of the decay rate dueto collective single-photon superradiance after an excita-tion of a timed Dicke state [10, 22]. This can be expectedsince by assuming an optically dilute cloud, we considerthe excitation of the cloud to remain driven mainly bythe laser, thus leading to the well-known superradiant be-havior of the cloud acting as a single large dipole. Here,we have re-derived this result for the switch-on evolutionof the intensity from the cloud.

Below, in Fig. 4(a), we demonstrate that the dampingparameters obtained from fits to CD simulations followthe predictions from a single-dipole limit only in the limitof very small “actual” optical thickness b(∆) 1, where

b(∆) =b0

1 + 4(∆/Γ)2. (19)

Note that generally the validity of the timed Dickestate approximation also requires negligible dephasingof the probe-beam across the sample, and thereforeb(∆)(∆/Γ) 1. In our setup, we expect to be outsideof this regime and that multiple modes will be involvedin the excitation dynamics.

2. Multiple modes

To analyze the mode structure it is convenient to

diagonalize the coupled-dipole matrix M in Eq. (8).This symmetric complex (not Hermitian) matrix can al-ways be diagonalized by an orthogonal transformation

AMAT = D with ATA = 1. Here, D is a diago-nal matrix containing the complex mode eigenvalues λη.In the transformed frame, the amplitudes of each mode

αη ≡ (Ab)η =∑mAη,mβm evolve as

αη(t) =wηλη

(eληt − 1

). (20)

The solution only depends on the complex eigenvalues,and the overlap of the uniform Rabi-frequency vectorwith the eigenmodes, wη = (Aw)η = iΩ

∑mAη,m.

The real and imaginary parts of each eigenvalue, λη ≡−Γη/2− iΩη, give rise to a damping and oscillation of therespective mode. As analyzed in [25] the mode popula-tion in the steady state, |αη(t→∞)|2 = |wη|2/|λη|2 anddepends on the geometrical factor, |wη|2, and the spectral

0 2 40

1

2

0 2 4 60

1

2(a)

0 5 100

1

0 5 100

1

(b)

Ne↵ = 5000<latexit sha1_base64="jk+R29+qIcrvsc4lxBPZtrIHqQU=">AAAB+3icbZDLSgMxGIUz9VbrbaxLN8EiuCqZquhGKLhxJRXsBdphyKSZNjTJDElGLMOAT+LGhSJufRF3vo3pZaGtBwIf5/whf06YcKYNQt9OYWV1bX2juFna2t7Z3XP3yy0dp4rQJol5rDoh1pQzSZuGGU47iaJYhJy2w9H1JG8/UKVZLO/NOKG+wAPJIkawsVbglm+DrKcEpFGUwyt4jhAK3AqqoqngMnhzqIC5GoH71evHJBVUGsKx1l0PJcbPsDKMcJqXeqmmCSYjPKBdixILqv1sunsOj63Th1Gs7JEGTt3fNzIstB6L0E4KbIZ6MZuY/2Xd1ESXfsZkkhoqyeyhKOXQxHBSBOwzRYnhYwuYKGZ3hWSIFSbG1lWyJXiLX16GVq3qnVZrd2eVOn6a1VEEh+AInAAPXIA6uAEN0AQEPIJn8ArenNx5cd6dj9lowZlXeAD+yPn8AcWVk3U=</latexit>

Ne↵ = 2000<latexit sha1_base64="8JAJaEyUQEGQvvHLTOWblZxlDoA=">AAAB+3icbZBLSwMxFIXv+Kz1Ndalm2ARXJVMFXQjFNy4kgr2Ae0wZNJMG5rMDElGLEPBX+LGhSJu/SPu/Demj4W2Hgh8nHNDbk6YCq4Nxt/Oyura+sZmYau4vbO7t+8elJo6yRRlDZqIRLVDopngMWsYbgRrp4oRGQrWCofXk7z1wJTmSXxvRinzJenHPOKUGGsFbuk2yLtKIhZFY3SFqhjjwC3jCp4KLYM3hzLMVQ/cr24voZlksaGCaN3xcGr8nCjDqWDjYjfTLCV0SPqsYzEmkmk/n+4+RifW6aEoUfbEBk3d3zdyIrUeydBOSmIGejGbmP9lncxEl37O4zQzLKazh6JMIJOgSRGoxxWjRowsEKq43RXRAVGEGltX0ZbgLX55GZrVindWqd6dl2vkaVZHAY7gGE7BgwuowQ3UoQEUHuEZXuHNGTsvzrvzMRtdceYVHsIfOZ8/wQCTcg==</latexit>

(c) (d)

FIG. 3. CD simulations for the parameters from Fig. 2. Be-sides the realization averaged curves (orange lines), we alsoshow 100 single realization results (thin grey lines). (a-c)Neff = 5000 (d) Neff = 2000. For small |∆| and large b0 weobserve large fluctuations that are independent on Neff .

factor 1/|λη|2. Furthermore, here it becomes evident thatthe respective mode occupations also depend on time. Itis interesting to note that at short times, at leading order(valid for t|λη| . 1), |αη(t → 0)|2 = |wη|2t2. This im-plies that for short times the population of the collectivemodes of the problem only depends on the geometricalfactor. The independence from the spectral factor canbe understood by the large frequency broadening of thedriving laser at the switch-on. This means that in anexperiment, the duration of the excitation pulse could beused to control the occupation of the different modes [21].

The time-dependent intensity signal, for the multi-mode case, can be written in the general form

I(t) =∑η,µ

αη(t)αµ(t)Gη,µ (21)

Gη,µ ≡∑n,m

A∗η,nAµ,mei(kn−k0)·(rn−rm). (22)

Here, from the geometrical contribution Gη,µ it becomesclear that if the measurement is in the forward direction(coherent scattering), cross-terms between modes playan important role, whereas if the measurement is off-axisand if an angle/realization average is considered (diffusescattering), different phases from different atoms averageto zero, and the dominating contribution to the intensitysignal stems from the diagonal mode populations |αη|2.

We find numerically that considering single atom po-sition realizations, the multi-mode structure can becomeclearly visible in the intensity signal. This is shown in

6

Fig. 3. There, for CD simulations, besides the aver-aged intensity signal from Fig. 2, we also show the in-tensity dynamics for 100 different realizations. For smallb(∆) ≈ 0.02 (b0 = 7.8, |∆| = 10) [Fig. 3(a)] there are onlysmall differences between position realizations, especiallyfor short times, and all realizations follow closely thesame curve that can be well fitted by Eq. (6) [Fig. 2(a)].In contrast, for larger b(∆), each realization exhibits verydifferent dynamics already at short times [Fig. 3(b-c)with b(∆) ≈ 0.2 and b(∆) ≈ 4.3, respectively]. Fur-thermore, those large fluctuations are also robust to thedifferent Neff that are accessible in simulations [comparepanels (c) and (d)]. We interpret this behavior as a sig-nature of multi-mode excitations, whose population andstructure fluctuate from one realization to another.

Remarkably, we find that the realization averaged sig-nal can still be decently modeled by just the effectivemode fitting function (6), although multiple modes areexcited. For example, as we have recently shown [19],for large b0 two frequencies stemming from a multi-modevacuum Rabi splitting play a crucial role in the dynam-ics, leading to an imperfect single-mode fit. Nevertheless,the fits still allow us to find effective mode properties ofthe cloud that are discussed in the next section.

E. Observation of superradiant damping

We now analyze the properties of the effective modeof the cold-atom cloud, and show that its excitation dy-namics indeed features superradiant decay as previouslyobserved in the switch-off in [5]. In Fig. 4 we summa-rize the scaling of the damping rate ΓN as a function ofthe laser detuning and optical thickness for both the CDsimulations Fig. 4(a) and the experiment Fig. 4(b). Theparameters are extracted from the fitting function (6) ofthe time-dependent intensity signal. For each fit we eval-uate the R2 value, and keep only points with R2 > 0.85.

As shown in Fig. 4, we find that indeed the collectivedamping shows a superradiant behavior, i.e. ΓN > 1.The behavior is very reminiscent of the results for theswitch-off dynamics obtained in [5], where two regimesof parameters can be identified. When the actual opti-cal thickness is very small, b(∆) 1, ΓN scales withthe resonant optical thickness b0 as it is predicted in thesingle-mode limit. This regime is highlighted in Fig. 4(a)by the comparison with Eq. (18) (solid lines). In the op-posite regime, b(∆) & 1, the superradiant rate presentsa reduction due to attenuation and multiple scattering,which can be attributed to an effectively reduced popula-tion of superradiant states close to resonance [25]. Thenb(∆) becomes the relevant scaling parameter, as seen bya collapse of the points onto a single curve when ΓN isplotted as a function of b(∆) [5]. This behavior seems toalso appear on the right edge in Fig. 4(a) and (b).

The observed experimental scaling in Fig. 4(b) is simi-lar to the full numerical CD simulation in Fig. 4(a), whichare realized using the same b0, yet very different densities

10−1 100 101

2

3

4

5

6

7

8

ΓN/Γ

(a) ∆ = −12Γ∆ = −10Γ∆ = −8Γ∆ = −7Γ∆ = −6Γ∆ = −5Γ∆ = −4Γ∆ = −3Γ∆ = −2Γ∆ = −1.5Γ

10−1 100 101

b(∆)

2

3

4

5

6

7

8

ΓN/Γ

(b) ∆ = −12Γ∆ = −10Γ∆ = −8Γ∆ = −7Γ∆ = −6Γ∆ = −5Γ∆ = −4Γ∆ = −3Γ∆ = −2Γ∆ = −1.5Γ

FIG. 4. Collective decay rate ΓN showing superradiant be-havior, reminiscent of previous switch-off experiments [5]. (a)Fits to CD simulations. (b) Fits to experimental data. Weonly kept fits for which the fitting function Eq. (6) works wellwith a value of R2 > 0.85. For small b(∆) the CD theoryagrees with the single-mode prediction [lines from Eq. (18)].Generally, the experiment exhibits a more damped behavior.

and atom numbers. However, especially for the under-damped case b(∆) . 1, we find a systematically largerΓN in the experiment than in the CD simulations. Mostrelevant dynamics occurs on a very short time scale, andwe thus attribute some of this systematic deviation tothe finite switch-on time in the experiment. We generallyfind that a more damped behavior at short times leadsto significantly larger estimations of ΓN in the effectivemode fit for those data points. Nevertheless, besides asystematic offset to larger values, the ΓN extracted fromthe experiment exhibit a similar scaling with b0 in theregime of b(∆) 1.

III. BEYOND THE LINEAR-OPTICS REGIME

While all results in the previous section were for a lowsaturation parameter in the experiment (s ≈ 0.02), we

7

now analyze the switch-on dynamics for larger satura-tion, beyond the linear-optics regime. We first provide asystematic derivation of a mean-field theory (Sec. III A).Then we show that the mean-field theory is capable ofsimulating experimentally observed switch-on dynamicsfor larger saturation (Sec. III B).

A. Theory beyond the linear-optics regime

1. Full quantum problem

Fully quantum mechanically, the system of N two-levelatoms is represented by a many-body density matrix ρwhich consists of a complex Hermitian 2N × 2N matrix.The non-relativistic dynamics of the system is describedby a quantum master equation (~ ≡ 1) [26–28]:

d

dtρ = −i[H, ρ] + L(ρ). (23)

Here, the first part describes coherent Hamiltonian dy-namics, i.e., the laser drive and exchange of excitations,

H = −∆∑n

σ+n σ−n +

1

2

∑n

(Ωnσ

+i + Ω∗nσ

−n

)+∑i 6=j

gij σ+i σ−j , (24)

where σ±n denote the standard spin raising and loweringoperators for a two-level atom at a position rn. Thesystem is considered in a frame oscillating at the laserfrequency (rotating wave approximation). The complexRabi frequency corresponds to the one defined in Sec. II Band the coherent coupling is gnm = ImGsnm. Note againthat throughout this paper we only consider the scalar-light model as a toy model, and thus neglect the near-fielddipole-dipole terms, but they can be easily included bymodifying the interaction kernel Gsnm [20].

The second term in Eq. (23) describes dissipative pro-cesses in the form of mutual decay, and has the generalform

L(ρ) =∑n,m

fnm(σ−n ρσ

+m − σ+

mσ−n , ρ

)(25)

=∑µ

γµ

(2LµρL

†µ − L†µLµ, ρ

). (26)

Here, ∗, ∗ denotes the anti-commutator, and the inco-herent decay rates are encoded in the symmetric matrixfnm = ReGsnm (including the n = m elements). In thesecond line we have written the dissipator in a standardLindblad form with the jump operators Lµ =

∑n unµσ

−n

that follow from a diagonalization of the real symmetricmatrix fnm =

∑µ γµunµumµ. The eigenvalues γµ deter-

mine whether the decay channel is super- or subradiant.Because of the exponential growth of the Hilbert space

with N , an exact time-dependent simulation of the full

quantum problem is computationally hard and currentlylimited to ∼ 20 atoms (using tricks such as quantumtrajectories [29]). The experimental setup is performed ina regime of small density and large optical thickness, forwhich simulation of much larger systems with N ∼ 103

are necessary. In the following we discuss how to reducethe complexity to tackle this regime.

2. Mean-field product state ansatz

To drastically reduce the size of the Hilbert space, acommon ansatz is to neglect any type of entanglementbetween the atoms. In such a mean-field (MF) situa-tion the full density matrix is considered to remain in aproduct state of the form

ρ =∏n

ρsn. (27)

Enforcing this factorized form at all times leads to time-dependent equations for each local density matrix ρsn

d

dtρsn = Trm6=n

(d

dtρ

). (28)

Due to the partial trace operation on the right hand sidethe MF equations of motion become nonlinear. Notethat the factorization property from Eq. (27) has alsobeen used in the Maxwell-Bloch description of a high-saturation regime in [18].

Every single-atom density matrix, ρsn, can be param-eterized by the complex expectations of βn = 〈σ−n 〉 andby zn = 〈σ+

n σ−n − σ−n σ+

n 〉 via ρsn = (1+2β∗nσ−n +2βnσ

+n +

znσzn)/2. Then, the Hamiltonian (24) and dissipator (25)

lead to the following compact form of the MF equations:

βn =

(i∆− Γ

2

)βn + iWnzn, (29)

zn = −Γ(1 + zn)− 4Im(βnW∗n). (30)

Here, we defined the general field acting on atom n as

Wn =Ωn2− i

∑m 6=n

Gsnmβm. (31)

Importantly the number of the nonlinear set of equationsin Eq. (29) and (30) only scales linearly with the systemsize, and thus a direct numerical integration is still feasi-ble for thousands of atoms. The physics behind the MFmodel is quite evident: all the atoms m create a meanfield that acts upon dipole n, Wn. The “coherent” realpart of Wn drives atom n just like the external laser,it comes from the virtual photon exchange in the dipole-dipole couplings. The “incoherent” imaginary part of Wn

gives rise to damping, and also to non-trivial evolution,which can lead to effects such as synchronization [30, 31].

The MF equations (29) and (30) become linear in thelow-excitation limit, as in the full quantum case. Then,

8

0 5 10 15

tΓ

0.0

0.5

1.0

I θ(a) kR = 1.0, s = 0.01

E

MF

CD

0 5 10 15

tΓ

0

1

2

(b) kR = 1.0, s = 2.0

0 5 10 15

tΓ

0

1

2

(c) kR = 5.0, s = 2.0

FIG. 5. Comparison between exact simulations of the master equation (E), Eq. (23), mean-field simulations (MF), Eqs. (29)–(30), and the coupled-dipole model (CD), Eq. (1). We use a small cloud with uniform density (Neff = 6). Here, ∆ = −4 and weuse two sphere radii kR = 1 (a-b) and kR = 5 (c). We tune the Rabi frequency to obtain small and large saturation parameters,s = 0.01 (a) and s = 2 (b-c), respectively. MF is superior to CD in reproducing the full quantum results for high saturation(c) and only fails for dense clouds and high saturation (b). Exclusion distance kdmin = 0.1, average over 21 realizations.

one can approximate zn ≈ −1 at all times, and recoverthe coupled-dipole equations

βn =

(i∆− Γ

2

)βn − iWn, (32)

which are identical to Eq. (1).In Fig. 5 we analyze the validity of the MF approxi-

mation by studying the switch-on dynamics for a smalltoy-model cloud consisting of Neff = 6 atoms. Here, weuniformely distribute the atoms in a sphere with differ-ent radii kR. For this small system we compare an exactsimulation of the master equation [Eq. (23)] to the pre-diction given by the MF product state ansatz [Eqs. (29)–(30)] and the coupled-dipole model [Eq. (1)]. Note thatfor numerical stability, in the toy-model setup of Fig. 5we still only consider far-field terms in the interactionkernel, although at such close distances near-field inter-action terms would play the dominant role.

For a small saturation parameter, here defined bys = 2Ω2

0/(4∆2 + Γ2), [s = 0.01 in Fig. 5(a)] we findthat, as expected, all three models agree with each other,even though the effective density is very high (kR = 1,sphere density ρs ∼ 1.4k3). For a larger saturation ofs = 2.0 in the high-density sphere, the linear coupled-dipole model provides very inaccurate results as seenin Fig. 5(b). Here, the MF result fails in capturing aslow decreasing slope for the intensity at later times, butstill reproduces the frequency of the oscillation reason-ably well. For a larger sphere (kR = 5, sphere densityρs ∼ 0.01k3) and high saturation of s = 2.0 we observethat MF can perfectly capture the exact intensity evolu-tion [Fig. 5(c)], while the CD simulation clearly fails.

The failure of MF can be attributed to the high den-sity, as for very closely spaced atoms large interactionsare inducing strong correlations between atoms beyondthe product state assumption. In contrast, for a rela-

tively dilute cloud for which the optical thickness canstill be high, we find that the MF assumption is validand the simulations also provide good estimations also inthe case of large saturation. Note that one can includefurther quantum corrections by including e.g. two-pointcorrelations in the equations of motions [15], thus also ac-counting for effects of entanglement between atoms. Suchcorrections, however, come at the price of increasing thecomplexity of simulations to ∼ N2

eff .

B. Comparison with the experiment beyond thelinear-optics regime

Finally, we compare experimental switch-on dynamicsin a high-saturation regime to simulations. The exper-imental data discussed in this section have been takenon the same apparatus as in Sec. II with a few upgradesdescribed in Refs. [8, 9]. However, the 10% to 90% risetime of the probe laser is now slightly longer, about 17 ns,because only acousto-optical modulators are used to pro-duce the pulses. This results in a more diffuse and slowon-set of the intensity signal. In comparisons with the-ory we compensate for this by shifting the time-signalsto match the first peak position. For all data points wefind good agreements with a shift of ∼ Γ−1, consistentamong the panels in Fig. 6.

In Fig. 6 we show results, at large saturation param-eters, and compare between experimental data and themean-field predictions, as well as the linear-optics CDsimulations. We show results for a large optical thick-ness of b0 ∼ 60 and for an increasingly large saturationparameter of s = 0.13, s = 0.34, and s = 0.63 in panelsFig. 6(a-c), respectively. It is striking that while the CDsimulations are capable of describing the experiment fora value of s ≈ 0.02 in Sec. II, here for s & 0.1 this linear

9

0 5 10

tΓ

0.0

0.5

1.0

I θ(a) b0 = 57.3, s = 0.13

exp.

MF

CD

0 5 10

tΓ

0.0

0.5

1.0

(b) b0 = 64.3, s = 0.34

0 5 10

tΓ

0.0

0.5

1.0

(c) b0 = 65.1, s = 0.63

FIG. 6. Comparison between the normalized intensity evolution from the experiment with MF and CD simulations in a largesaturation regime (s & 0.1) and with b0 ∼ 60: (a) s = 0.13, (b) s = 0.34, (c) s = 0.63. Here, ∆ = −4Γ. For the simulationsNeff = 5000 and results for 78 random atom positions have been averaged.

model is insufficient. Moreover with increasing s, we ob-serve that the CD prediction (which does not depend on sdue to the linearity of the CD equations) becomes worse.The MF simulations, in contrast, predict the trend ofthe experimental data of exhibiting a more pronouncedoscillation with increasing s.

IV. CONCLUSION & OUTLOOK

We have demonstrated that the time evolution of theintensity of laser light scattered off a cold-atom cloud canbe used to observe collective effects, in particular super-radiance. Here we have shown that superradiance cannot only be observed after the laser is rapidly switchedoff, as in [5], but also in the damping of oscillations im-mediately after the laser is switched on.

In a limit of low intensity/saturation, the dynamicscan be described by a linear-optics coupled-dipole model,which matches the experimental behavior very well. Forlow optical thickness, the results for superradiant damp-ing follow the predictions of a single “mean” mode ap-proximation (timed Dicke state excitation). In general,and especially for larger optical thickness, multiple modesare excited. Nevertheless, the cloud can still be reason-ably well modeled by an effective single-mode response.

We furthermore showed that when the saturation is in-creased, the coupled-dipole model becomes insufficient,as expected. Instead, for this regime we find that ex-perimental data can be well described by a simulation ofnon-linear mean-field equations that follow from a prod-uct state ansatz. We showed that this efficient numericalapproach works well for simulating dynamics in diluteclouds with large excitation fractions and large opticalthickness.

It will be interesting to analyze signatures going be-yond the coupled-dipole and mean-field assumptions dis-cussed here, which has been a topic of recent inter-

est [13–17]. Here, in particular we showed that the time-dependent switch-on intensity signal does not discernsuch signatures unless going to a high-density regime.There the time-dependent response could be a useful toolfor quantum signatures [18, 23, 32–34]. Beyond mean-field corrections could be included using approaches thattake quantum correlations between pairs of atoms intoaccount [15], by exploiting a semi-classical phase spaceapproach [35] or a combination of both [36].

V. ACKNOWLEDGEMENTS

We thank Ivor Kresic and Michelle Araujo for theircontribution in setting up the fast switch-on sys-tem and Luis Orozco for fruitful discussions. Partof this work was performed in the framework ofthe European Training Network ColOpt, which isfunded by the European Union (EU) Horizon 2020programme under the Marie Sklodowska-Curie action,grant agreement No. 721465. R.B. and T.S.E.S. ben-efited from Grants from Sao Paulo Research Founda-tion (FAPESP) (Grants Nrs. 2018/01447-2, 2018/15554-5 and 2019/02071-9) and from the National Councilfor Scientific and Technological Development (CNPq)Grant Nrs. 302981/2017-9 and 409946/2018-4. R.B. andR.K. received support from project CAPES-COFECUB(Ph879-17/CAPES 88887.130197/2017-01). P.W. is sup-ported by the Deutsche Forschungsgemeinschaft (grantWE 6356/1-1). J.S. is supported by the FrenchNational Research Agency (ANR) through the Pro-gramme d’Investissement d’Avenir under contract ANR-11-LABX-0058 NIE within the Investissement d’Avenirprogram ANR-10-IDEX-0002-02. Research was carriedout using computational resources of the Center forMathematical Sciences Applied to Industry (CeMEAI)funded by FAPESP (grant 2013/07375-0) and the Cen-tre de calcul de l’Universite de Strasbourg.

10

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